Geodesicity and isoclinity properties for the tangent bundle of the Heisenberg manifold with Sasaki metric

We prove that the horizontal and vertical distributions of the tangent bundle with the Sasaki metric are isocline, the distributions given by the kernels of the horizontal and vertical lifts of the contact form w on the Heisenberg manifold (H3,g) to (TH3,gS) are not totally geodesic, and the distributions FH=L(E1H,E2H) and FV=L(E1V,E2V) are totally geodesic, but they are not isocline. We obtain that the horizontal and natural lifts of the curves from the Heisenberg manifold (H3,g), are geodesics on the tangent bundle endowed with the Sasaki metric (TH3,gs), if and only if the curves considered on the base manifold are geodesics. Then, we get two particular examples of geodesics on (TH3,gs), which are not horizontal or natural lifts of geodesics from the base manifold (H3,g).

Anahtar Kelimeler:

Tangent bundle, Sasaki metric, Heisenberg metric, geodesics

Geodesicity and isoclinity properties for the tangent bundle of the Heisenberg manifold with Sasaki metric

Keywords:

Tangent bundle, Sasaki metric, Heisenberg metric, geodesics,

PDF

___

[1] Abbassi, M.T.K., Sarih, M.: On some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds. Diff. Geom. And its Appl. 22 , 19-47 (2005).
[2] Anastasiei, M., Vacaru, S.: Fedosov quantization of Lagrange-Finsler and Hamilton-Cartan spaces and Einstein gravity lifts on (co) tangent bundles. J. Math. Phys. 50, 1, 013510, 23 pp. (2009).
[3] Blair, D.E.: Contact manifolds in Riemannian Geometry. Lectures Notes in Math. n. 509, (1975).
[4] Boothby, W.: An introduction to differentiable manifolds and Riemannian geometry. Academic Press, New York, (1975).
[5] Caddeo, R.; Gray, A.: Curve e Superfici. Ed. CUEC 2000.
[6] Do Carmo, M. P.: Riemannian geometry. Birkhauser, Basel (1992).
[7] Duˇsek, Z.: On the reparametrization on affine homogeneous geodesics. Differential geometry, Proceedings of the VIII International Colloquium, Santiago de Compostela, Spain, 7-11 July 2008, Jes´us A. Alvarez Lopez, Eduardo ´ Garc´ıa-R´ıo (eds), World Scientific, 217-226 (2009).
[8] Duˇsek, Z.: Structure of geodesics in a 13-dimensional group of Heisenberg type. Proc. Colloquium on Differential Geometry, Debrecen, 95-103 (2001).
[9] Duˇsek, Z., Kowalski, O., Vl´aˇsek, Z.: Homogeneous geodesics in homogeneous affine manifolds. Rezult. Math., 54/3-4, 273-288 (2009).
[10] Gheorghiev, Gh., Oproiu, V.: Variet˘at¸i diferent¸iabile finit ¸si infinit dimensionale. Vol. 2, Editura Academiei Romˆane (1978).
[11] Gudmundsson, S.; Kappos, E.: On the geometry of tangent bundles, Expo. Math. 20, p.1-41,(2002).
[12] Hangan, T.: Sur les transformations g´eom´etriques de l’espace de Heisenberg. Rendic. Semin. Facolt`a di Scienze M.F.N. Univ. di Cagliari (1985).
[13] Jany ˇska, J.: Natural vector fields and 2-vector fields on the tangent bundle of a pseudo-Riemannian manifold. Archivum Mathematicum (Brno) Tomus 37, 143-160 (2001).
[14] Kol´aˇr, I., Michor, P., Slovak, J.: Natural Operations in Differential Geometry. Springer Verlag, Berlin, vi, 434 pp (1993).
[15] Kowalski, O., Sekizawa, M.: Natural transformations of Riemannian metrics on manifolds to metrics on tangent bundles-a classification. Bull. Tokyo Gakugei Univ. (4), 40, 1-29 (1988).
[16] Lutz, R.: Quelques remarques sur la g´eom´etrie m´etrique des structures de contact. Coll. Travaux en cours Hermann, Paris, (1984).
[17] Miron, R.: On the notion of Lagrangian, Finsler, Hamiltonian and Cartan Mechanical Systems. In: R.M. Santilli (ed), Proceedings of the XVIII Workshop on Hadronic Mechanics, (2006).
[18] Miron, R., Anastasiei, M., Bucataru, I.: The Geometry of Lagrange Spaces. In: Antonelli, P.L. (ed), handbook of Finsler Geometry, vol. II, Kluwer Academic Publishers, 969-1124, (2004).
[19] Munteanu, M.I.: Cheeger Gromoll type metrics on the tangent bundle. Proceedings of Fifth International Symposium BioMathsPhys, Iasi , June 2006, U.A.S.V.M. Ion Ionescu de la Brad, 49, 2, 257-268 (2006).
[20] Munteanu, M.I.: Old and New Structures on the Tangent Bundle. Proceedings of the Eighth International Conference on Geometry, Integrability and Quantization, June 9-14, 2006, Varna, Bulgaria, Ed. I. M. Mladenov and M. de Leon, Sofia, 264-278 (2007).
[21] Munteanu, M.I.: Some Aspects on the Geometry of the Tangent Bundles and Tangent Sphere Bundles of a Riemannian Manifold. Mediterranean Journal of Mathematics, 5 , 1, 43-59 (2008).
[22] Nagy, P.T.: Geodesics on the tangent sphere bundle of a Riemannian manifold. Geom. Dedicata, 7, 233-243 (1978).
[23] Oproiu, V.: A generalization of natural almost Hermitian structures on the tangent bundles. Math. J. Toyama Univ., 22, 1-14 (1999).
[24] Oproiu, V.: Some new geometric structures on the tangent bundles. Publ. Math. Debrecen, 55/3-4, 261-281 (1999).
[25] Oproiu, V., Papaghiuc, N.: General natural Einstein K¨ahler structures on tangent bundles. Differential Geometry and Applications, 27, 3, 384-392 (2009).
[26] Piu, M.P.: Sur certains types de distributions non-integrables totalement g´eod´esiques. Th`ese de Doctorat, Univ. de Haute Alsace, 116 pp. (1988).
[27] Piu, P., Profir, M.M.: On the geodesics of the rotational surfaces in the Bianchi-Cartan-Vranceanu spaces. Differential geometry, Proceedings of the VIII International Colloquium, Santiago de Compostela, Spain, 7-11 July 2008, Jes´us A. Alvarez Lopez, Eduardo Garc´ ´ ıa-R´ıo (eds), World Scientific, 306-310 (2009).
[28] Salimov, A. A., Gezer, A., Akbulut, K.: Geodesics of Sasakian Metrics on Tensor Bundles. Mediterranean J. Math., 6, 2, 135-147 (2009).
[29] Salimov, A. A., Kazimova, S.: Geodesics of the Cheeger-Gromoll Metric. Turk J. Math., 33, 99-105 (2009).
[30] Sasaki, S.: On the differential geometry of the tangent bundle of Riemannian manifolds. Tˆohoku Math. J., 10, 238-354 (1958).
[31] Tahara, M., Vanhecke, L., Watanabe, Y.: New structures on tangent bundles. Note di Matematica (Lecce), 18, 131-141 (1998)
[32] Wood J.C. : Harmonic morphisms, foliations and Gauss maps, Contemporary Ma thematics, 49 (1986), 145-184.
[33] Yano, K., Ishihara, S.: Tangent and Cotangent Bundles. M. Dekker Inc., New York (1973).